Chapter II: Part 2

Michelson and Morley worked on the theory that if they sent rays of light from west to east (the direction of the earth’s rotation) and then reflected them back over their course it should take longer to make the eastward trip than the westward, because in the first instance the earth is carrying the objective point away from the light while in the latter instance it would be rushing to meet the oncoming reflected rays. Under this condition there should be a noticeable _interference_ of the light waves due to the difference in distance and time involved in making the two halves of the round trip. But to the amazement of all there was no interference whatever, notwithstanding the fact that the apparatus was ten times larger than it needed to be to reveal such interference of the waves had it really occurred.

The conclusion reached by Einstein as a result of this experiment is that since light rays consist of matter in its basic or electronic state, freed from atomic attraction, they therefore possess the limiting velocity of which matter is capable. Hence they could not travel more rapidly than 186,300 miles a second even if given a quick send-off, nor would our traveling toward the light affect its apparent velocity to us--unless it were possible for us to be traveling forward more rapidly than light itself can travel. This would undoubtedly be impossible, inasmuch as any physical body would necessarily consist of electrons in the atomic or “bound” state and therefore could not possess the mobility that free electrons would enjoy. The universe, then, being a four-dimensional continuum, is so constructed that the velocity of light always appears constant to all observers within it.

This is what Einstein means when he postulates that light _in vacuuo_ (i. e., unobstructed) possesses a constant velocity irrespective of the relative velocity of observer and source of light. That is, it is constant so far as the observer is concerned. Thus if a flash should occur on any heavenly body and we were moving toward the flash at say 40,000 miles a second and another observer were moving away from it at say 60,000 miles a second, the experiment of each observer would indicate that the light has reached him at exactly 186,300 miles a second, although according to Euclid’s conception of space the light has been obliged to travel 100,000 miles a second faster to reach the one observer than the other. But Euclid’s conception is faulty, as will be seen shortly.

How, then, would it be possible for the light rays to possess the same apparent velocity per second for the two observers? It would not be possible if “time” and “distance” are absolute quantities having the same meaning for all observers. But if “seconds” and “miles” mean one thing to observer “A” and a totally different thing to observer “B,” then the apparent contradiction of facts becomes harmonious. This is the essence of the doctrine of relativity. Observer “A” himself does not use the terms “seconds” and “miles” consistently, i. e., as unvarying quantities, nor does anyone. They mean one thing today and something else tomorrow, depending upon what we are measuring and the relative velocity between the observer and the object. The observer is not aware of this inconsistency. To him there is no inconsistency whatever. Nevertheless, only by acknowledging the varying quantities of time and of space, and admitting the geometry which combines the two into one unit, can the Michelson-Morley experiment and other similar observations be understood and explained.

=Simultaneity a Meaningless Term=

We have been taught that the true length of a moving body is “the distance between _simultaneous_ positions of its end points”--a very good definition, but impossible of application for the reason that we cannot determine the _simultaneous_ positions of any two points in the universe. Simultaneity is a meaningless term so long as the absolute velocity of the observer and the absolute velocity of the object being measured are unknown. We may know the _relative_ velocity between them, but that is not sufficient. The two may be relatively at rest--but for all we know the entire universe may be speeding through space at thousands of miles a second in either one direction or another.

We may _see_ two events occur at the same instant, but that does not prove that they actually _occurred_ simultaneously. Before we could compute the exact time of the occurrence of either of the events we must know the direction in which, and the velocity at which the universe as a whole is moving, together with any and all velocities of the observer at the moment. This knowledge we do not possess. Until the _absolute_ velocity of bodies can be determined the question of simultaneity must remain unsolved.

=The General Principle of Relativity=

When in 1905 Einstein published the foregoing postulates which are limited to uniform, rectilinear motion he may have considered that it would be expecting too much to look for a general principle of relativity such as would hold good for all kinds of rotating and irregular motions and by which observers of different and variable velocities might agree as to the reality of things under their observation. Concluding, however, that the universe must surely be constructed in a consistent manner he finally set out to find some rule or principle by means of which an observer in one region would be seen to possess no advantage over an observer in any other region of the great expanse in arriving at accurate conclusions.

Of course Einstein hardly expects to go to the Pleiades or to Betelguese and from there take measurements and make calculations; he is doubtless content to make all his observations from this earth. But how may he be sure that observations made from a reference frame located in this particular region of the universe will be true to the reality since it is manifest that observers located elsewhere and using different reference frames must necessarily reach conclusions different from ours if they employed our accustomed laws? Maybe they would be much nearer the reality than we! What right have _we_ to assume a monopoly on truth! None whatever until we can formulate nature’s laws in a manner that will hold good for every part of the universe alike.

Until we are able to do this our science must be like the vain efforts of the unskilled fisherman who harpoons for fish. Ignorant of the trick that water plays on the line of sight he strikes directly at the spot where he “sees” the fish and always misses his prey. The skilled harpooner, on the contrary, understands the law of refraction of light rays in water, and knows how to allow for this refraction; hence he strikes a little this side of where the fish appears to be and is rewarded with success. He is guided by a proven law and thereby ascertains the true location of the fish, whereas the other man follows “blind” observation which is quite frequently deceptive.

Einstein’s “General Principle of Relativity” is not, in fact, a mere generalization of the Special Theory in the sense that it simply enlarges upon the two postulates which we have already considered. On the contrary it handles the subject of Relativity from quite a new standpoint, and therefore might be said to belong to an entirely different school of thought. It does not lend itself to visualization as readily as does the Special Theory, and is consequently more difficult of explanation and comprehension. However, what we have already learned concerning Relativity will materially aid us in understanding what follows, for the two theories are, after all, dealing with the same general subject matter. We shall therefore endeavor to link the two phases of the subject in a logical and consistent manner.

We know, as a matter of fact, that “uniform, straight-ahead motion” which Einstein in his original theory assumed to exist, is an ideality that does not appear in nature, because all motion with which we are familiar is to some extent irregular, nor does any material object move in a perfectly straight line. But realizing the necessity for a standard from which to proceed, Einstein properly enough assumed a standard of absolute perfection and absolute simplicity of motion, even though it does not actually exist anywhere around us. In exactly the same manner Euclidean geometry assumes and deals with theoretical points, lines and planes which have no material existence in fact.

As set forth in Einstein’s first postulate of the Special Theory, an observer on a uniformly moving system could not possibly detect the motion of his system without making reference to some outside object. In the case of bodies or systems moving irregularly (i. e., with acceleration) however, an observer thereon would detect “forces” acting upon himself and upon all other objects on his system, due, of course, to the acceleration. Recalling the illustration of the moving train: so long as it is moving with perfect uniformity an observer thereon would not know he is in motion at all until he made a comparison with some outside object. But if the train suddenly slows down he is thrown forward in his seat; if it speeds up he is thrown backward. This force is called _inertia_. Now if we had never experienced it before and were put aboard a noiseless and uniformly moving car from which we could not see out we would be unable to interpret these strange “forces” that we would feel as the motion of the car became accelerated. We would probably attempt to explain them as some sort of magnetic attraction, exactly as we are accustomed to explain the “force” of gravity.

=Gravitation and Inertia=

In the General Principle of Relativity Einstein deals with these forces (inertial and gravitational) and attributes them to a common cause, viz., acceleration of motion, and has put the matter upon a consistent mathematical basis which at once accounts for certain discrepancies long observed in Euclidean geometry and in Newton’s laws. It is obvious enough that where there is no acceleration of motion there could be no centrifugal or inertial force exhibited: but we have been accustomed to looking upon gravitation as something entirely different--as a mysterious drawing power or attractive force that is somehow inherent in matter. But gravitation is non-existent if we fall with the proper acceleration. To use Einstein’s own illustration: if we were in a closed room poised somewhere in gravitational space, and began to fall with the acceleration common to that field, there would be no gravitational effects to be observed. Objects released by our hand would not fall but would remain where they are, and we could raise ourselves from the floor and stand midway between the floor and the ceiling as easily as upon the floor itself.

Again assume we are in a closed room poised in space, in a region remote from any gravitational field whatsoever. Then suppose we began to rise with a constant acceleration. Forthwith we would feel our feet pressing against the floor. Objects released from our hand would strike the floor by reason of the floor rising up to meet them, and in all respects the effects would be identical with that of gravitation. In other words we would have created an artificial gravitational field, and it would be due to our accelerated motion.

The characteristics of gravitation and inertia are identical. No amount of insulation or screening will diminish the “pull” of gravity on anything. Furthermore, gravity acts on every kind and quantity of matter alike, so that if a feather weighing less than an ounce and a pig of lead weighing a ton were held side by side at the top of a great vacuum tube and allowed to drop at the same instant, the feather would reach bottom within the same time as the lead, each falling at an acceleration of approximately 32 feet per second. It is the resistance of the air that retards the fall of light materials, such as a feather, but in a vacuum there is no resistance and gravity is found to act on all matter to the same degree under such conditions. The same is true of inertia _in vacuuo_.

When this relationship between the two forces is recognized we are prepared to believe Einstein when he states that inertial force and gravitational force are due to a common cause, viz., acceleration. This does not mean that our earth, for instance, is being accelerated in all directions at once, expanding out to meet “falling” objects such as in the case of the artificial gravitational field mentioned in the above paragraph. It does mean, however, that the falling objects themselves are accelerated, but as will be presently seen this acceleration is not due to any attractive force exerted by a “center of gravity” but rather to a warped condition of space which surrounds all bodies of matter.

Neither Newton nor Einstein have attempted to analyze the structure of matter and on this basis explain the phenomenon of gravitation. Newton evidently believed, however, that every particle of matter exerts a drawing force upon every other particle of matter, hence he formulated his law which specifies this attraction between bodies as being directly proportional to the product of their mass and inversely proportional to the square of the distance between them. But he did not attempt to make clear what that “drawing force” is, or _why_ it is inherent in all matter, nor did he explain how or through what medium or mechanism it operates.

Newton contented himself with merely dealing with the phenomenon of gravitation in the abstract. So does Einstein, but with this difference: the latter denies the existence of any mechanism whatever in connection with gravitational force so far as any attractive power from within is concerned, and accounts for it on purely geometrical grounds. This is the most difficult phase of the Einstein theory for the layman to grasp, for the reason that it involves the whole structure of non-Euclidean geometry with which the public is generally unfamiliar.

=Non-Euclidean Geometry=

Euclid, the famous Greek mathematician, in the third century B. C. published the first systematic treatise on geometry (the science of space and its measurement), and his axioms and theorems are generally taught in our high schools and colleges today. Euclid proceeded upon the simple theory that all space consists of points, lines and planes. He defined a _point_ as that which has position but not size; a _line_ (continuity of points) as possessing length but no breadth or thickness; and a _plane_ (continuity of lines) as having length and breadth, but no thickness. They are simply abstract terms having no physical existence in nature, except as they exist in our minds. Nevertheless they have proved themselves convenient in measurement and calculation.

But when mathematicians, after centuries of earnest effort, were unable to prove Euclid’s postulate concerning parallel lines, it occurred to some of them that possibly the whole Euclidean system rests upon a faulty foundation. Then it was that Saccheri in Italy, Legendre in France, Gauss in Germany, Bolyai in Hungary and Lobatschewsky in Russia, all masters of Euclidean geometry, conceived of other methods of decomposing space than that proposed in Euclid’s _Elements_.

Thus it was that early in the nineteenth century, almost simultaneously in many countries, did many non-Euclidean geometric works come to be published. These were of the same general character or form, commonly called Hyperbolic geometry. Each of them is as consistent in itself as is the geometry of Euclid. But to Riemann belongs the credit of formulating a geometry which in the light of Einstein is seen to approach much nearer to the reality of nature than does the Euclidean or any other system.

Riemann produced his general work along this line in 1854 which was far ahead of his time. He actually prophesied the connection of geometry with matter, and had he possessed a little more vision he would doubtless have worked out the details as well as the principles underlying gravitation in much the same manner as Einstein has done. Riemann’s efforts in the field of non-Euclidean geometry has materially aided Einstein in the development of the present theory. Minkowski’s work was utilized by Einstein to much profit in the outworking of the Special Theory, particularly his clarification of _time_ as a fourth dimension.

=Time as a Fourth Dimension=

It is natural for us to think of all matter as possessing but three dimensions--length, breadth and thickness--and we have been accustomed to making our measurements of matter and of space on that basis. Using the formula of Pythagoras we have ascertained the distance between any two points in a _plane_ (a two-dimensional area) by extracting the square root of the sum of the squares of the co-ordinate axes, i. e., the base and the altitude as in the accompanying diagram. See Figure 4.

If point A is 8 miles south and 6 miles west of point B then A and B are 10 miles apart, thus:

The square of 8 is 64 The square of 6 is 36 --- The square root of 100 is 10

Likewise, the distance between any two points in a three-dimensional region (as from an upper to the remotest lower corner of a room) is generally considered to be “the square root of the sum of the squares of the three sides” (Fig. 5).

Thus if the distance O to X is 12 feet and X to Y is also 12 feet, while Y to Z is 14 feet, then the straight diagonal distance from O directly through the room to point Z is 22 feet, because the sum of the squares of the three sides (144 + 144 + 196) yields a total of 484, and the square root of that number is 22. This simple formula will hold good for all ordinary measurements, but for great distances in space a slight correction is found necessary because of the little trick that light rays are prone to factor, i. e., the numerical value of the interval of time required for a light ray to traverse play upon us. We must subtract the _time_ from the distance. Hence if our cube were large enough to fill a goodly portion of the universe we would no longer say that the diagonal distance from O to Z is √(x^2 + y^2 + z^2) but rather √(x^2 + y^2 + z^2 - t^2), ----t, of course, representing _time_.

Now recall what we learned in the preceding pages, that the velocity of light always appears to be the same to all observers irrespective of the relative velocity between the observer and the source of light. It is manifest, therefore, that in making measurements the time factor (t) _really_ represents one quantity for one observer and a totally different quantity for another observer notwithstanding the fact that it appears to be a constant to all observers. Inasmuch as the velocity of light does appear to be constant to all observers its actual stretching or contracting of units is not manifest. Therefore the corrected equation as given above (the subtraction of the _time_ element) holds good for all observers irrespective of their motion.

The point of interest to the non-Euclidean geometer in connection with any measurement, be it remembered, is not the abstract _distance_ between points, because distance is not a constant and is not determinable unless we know the _absolute_ velocity of the observer and of the points being measured, which knowledge we do not possess. What we should look for, then, is the _distance and time combined_, or the _separation-interval_ as it is aptly called. The time factor automatically corrects the units for each observer, no matter what his motion may be, and thus the separation-interval appears a constant.

The foregoing illustrates how time takes its place alongside the ordinary three dimensions of space, and is in reality a fourth dimension, although it is not a thing that can be visualized as we can visualize the length or breadth or thickness of any object. In the following paragraphs we shall examine further into the geometry of the universe, particularly as it relates to the phenomenon of gravitation.

=Geometry with a Physical Meaning=

Certain news dispatches and book reviews have erroneously reported Professor Einstein as having said “only twelve men in the world can understand the Principle of Relativity.” The statement becomes absurd in view of the scores of volumes now in print, all of which set forth more or less clearly the details of the Einstein theory. What he alluded to in the remark so generally misquoted and misconstrued is his mathematical equations (calculus of tensors). He questioned if there are more than a dozen mathematicians in the world who are familiar with this abstruse differential calculus because it is not generally taught in the university text books.

This calculus is a veritable maze of formulæ, really invented by Riemann and Cristoffel, but systematized by the celebrated Italian mathematicians, Ricci and Levi-Cevita, and is impractical for any ordinary use. This is why so few mathematicians have familiarized themselves with it. Einstein, however, found it invaluable in dealing with such complex geometrical problems as his theory produced.

Briefly, the non-Euclidean geometer deals with _surfaces_ rather than planes, and his fundamental postulates are sufficiently broad to apply to all regular surfaces whether they be planes, spheres, cylinders, conicoids or even spheroids or ellipsoids. He considers a “straight” line as being the shortest distance between two points _on a surface_, hence if the surface is curved the “straight” line connecting any two points thereon will also be curved. This _shortest_ distance between points is called a _geodesic_. If the surface happens to be a _plane_ then the geodesics connecting points thereon are really straight lines in the Euclidean sense, but this would not be true for any other kind of a surface. Thus it is seen that Euclidean geometry is simply a limiting case of this more general geometry.

Geometers of the elliptic or spherical school, including Einstein, declare that in nature there is no such thing as a purely Euclidean straight line such as may be prolonged in opposite directions to infinity. On the contrary they hold that any “straight” line if prolonged sufficiently would return upon itself, because the universe is so constructed. In other words, what we ordinarily call a straight line is but an arc of a near infinite circle which possesses the _least possible curvature_. Magazine writers in an endeavor to make clear this portion of the theory of Relativity have strikingly declared that “according to Einstein a man might look through a telescope in any direction whatsoever and behold the back of his neck.” This jest, though omitting essential facts, is not without geometrical foundation. If we possessed a near infinite telescope and should live for a near infinite period of time to enable the rays of light to traverse this near infinite circle, then, if there were no obstructions along our line of sight, we might be rewarded with a round trip view of the rear portion of our body--though the simpler method would be to use two ordinary mirrors.

All this, however, has an important bearing upon Einstein’s interpretation of gravitation. Not only does he contend for Lobatschewsky’s “curvature of space” but he also holds that surrounding every body of matter there is a _special_ space-curvature (four dimensional), the degree of which depends upon the body’s observed mass. This special curvature or “warp” of space constitutes the “gravitational field” surrounding all large bodies of matter and causes the acceleration of falling particles in that field. This distortion of space increases in proportion to the mass of the body causing it, and decreases with the distance from that body until ultimately it becomes _nil_ or practically so in a region remote from all matter.

Perhaps the nearest approach to a visualization of this space-curvature (which constitutes a gravitational field) is to consider the lines of force in a magnetic field. The reader is doubtless familiar with the age-old experiment of placing file dust on a thin sheet of cardboard or plate of glass and then holding a horseshoe magnet underneath with the two poles touching the sheet or plate. Immediately the filings arrange themselves into curved lines between the poles as shown in Fig. 6.

This experiment indicates that between the poles of a magnet are constant lines of force, invisible to sight but manifesting themselves when attractable particles are in or near their path. The earth, likewise, is a great magnet, having one magnetic pole in upper Canada above Hudson Bay, about 70° north latitude, and another pole in the Antarctic Ocean south of Australia. Between these two magnetic poles continually flow these invisible curved lines of force, just as with our horseshoe magnet in Figure 6. These lines of force are the cause of compasses pointing in a northerly and southerly direction. Other planets undoubtedly possess magnetic poles similar to those of the earth.

=The Background of Gravitation=

Now let us conceive of invisible geometric lines pervading the entire spatial universe somewhat analogous to these lines of force in a local magnetic field. To each point in these lines let us ascribe an electric and gravitational potential (remembering, of course, that we are dealing with _four_-dimensional space), and we have before us, in a nearly visualized, sense, the background of the new theory of gravitation.

Einstein was the first to present the subject of gravitation from this viewpoint. For centuries up to this time geometry and physics were considered as belonging to entirely different schools of thought, but under the master hand of Einstein the two sciences have been welded together into one. As Freundlich puts it, “quantities which hitherto had only a purely geometrical import, for the first time became animated with physical meaning.” Thus “empty space” is no longer empty, even though the existence of the ether be denied. When the study of free electrons has sufficiently advanced, it may be seen that these elementary particles of electricity, or energy-particles, freed from atomic or mass attraction, play an important role in gravitational phenomena.

Figure 7 represents in a crude fashion the special curvature of space in the region of a large body of matter, for instance our earth, with the points (events) situated at finite instead of infinite nearness to each other for sake of illustration. It will be readily seen that the distortion of the geometric lines would necessarily alter the relative positions of the point-potentials.

Any falling body moves in a geodesic, i. e., from one point to the _next nearest_ point in space-time.[2] In an undisturbed region, remote from matter, the points (events) may be considered as so arranging themselves that any four neighboring ones would constitute practically a square. It may then be seen that the easiest path for falling bodies would be to follow the _sides_ of the squares because by so doing they would be following the _geodesic_ or shortest distance between points. (See Fig. 7.) But in the region of a large body of matter the lines of points become so distorted that the _diagonal_ of any four neighbor points becomes the geodesic. Then the path of the falling particle will accordingly deviate. It will always follow the geodesic, or easiest path.

This causes the falling particle to take a direction which points toward the center of the gravitational field--but the center of gravity is exerting no drawing or attractive force as Newton supposed. Gravity is thus seen to be not an external _drawing_ power operating between bodies of matter, but an inherent order of nature in space. The acceleration as well as the direction of the falling particle is accounted for by this theory. As the separation-interval between points becomes shorter--due to the constantly accentuated distortion as the large body is approached--the falling particle would be correspondingly accelerated. The distortion being constantly increased the acceleration would likewise be constant.

Newton, in his law of inertia, postulated that any particle of matter at rest will forever remain at rest if not disturbed, but when once set in motion it will continue to move at uniform velocity in a straight line unless interfered with by outside force. Einstein, on the contrary, holds that any particle of matter if left to itself will _move_ (let us say _fall_, if you please) in the easiest direction (i. e., in a geodesic) at constant velocity unless it encounters a gravitational field (a distorted region), in which event it will become accelerated, and will also, if necessary, change its direction, in obedience to the principle of least action. In other words, it is natural for matter to possess energy, therefore natural for it to be in motion and unnatural for it to be at rest. And the contention has this much in its favor: every particle of matter in the universe, from the infinitesimal electron to the more gigantic sun and super-system of outer space, _is moving_, so far as our most modern observations extend. Nothing has yet been discovered to be at rest.

[Footnote 1: When either “space” or “time” is mentioned independently of the other in this treatise it may be understood that the terms are used in the ordinary conceptual sense for purposes of simplicity.]

[Footnote 2: The reader must bear in mind that four-dimensional, not the ordinary three-dimensional, space is here discussed. The author has endeavored, however, to treat the matter in such a manner as to approach a visualization of this otherwise quite complex subject.]

=TRANSCRIBER’S NOTES=

Simple typographical errors have been silently corrected; unbalanced quotation marks were remedied when the change was obvious, and otherwise left unbalanced.

Punctuation, hyphenation, and spelling were made consistent when a predominant preference was found in the original book; otherwise they were not changed.